Given a group G, we can define Gop to be a new group whose underlying set is the same as that of G but with the new multiplication g.h = hg, i.e. multiply as if you were in G but reverse the order.
Is there anything interesting to say about this construction?
When is a group isomorphic to its opposite group?
One sees the "opposite ring" of a non-commutative ring showing up in a crucial way in number theory in the study of Brauer groups, so this is not necessarily an artificial question. This is just a question that has been poking around my brain for a little while. . . it seems like a potentially neat little problem.